Classical plume theory: 1937-2010 and beyond

Developing a theoretical description of turbulent plumes, the likes of which may be seen rising above industrial chimneys, is a daunting thought. Plumes are ubiquitous on a wide range of scales in both the natural and the man-made environments. Examples that immediately come to mind are the vapour plumes above industrial smoke stacks or the ash plumes forming particle-laden clouds above an erupting volcano. However, plumes also occur where they are less visually apparent, such as the rising stream of warmair above a domestic radiator, of oil from a subsea blowout or, at a larger scale, of air above the so-called urban heat island. In many instances, not only the plume itself is of interest but also its influence on the environment as a whole through the process of entrainment. Zeldovich (1937, The asymptotic laws of freely-ascending convective flows. Zh. Eksp. Teor. Fiz., 7, 1463-1465 (in Russian)), Batchelor (1954, Heat convection and buoyancy effects in fluids. Q. J. R. Meteor. Soc., 80, 339-358) and Morton et al. (1956, Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A, 234, 1-23) laid the foundations for classical plume theory, a theoretical description that is elegant in its simplicity and yet encapsulates the complex turbulent engulfment of ambient fluid into the plume. Testament to the insight and approach developed in these early models of plumes is that the essential theory remains unchanged and is widely applied today. We describe the foundations of plume theory and link the theoretical developments with the measurements made in experiments necessary to close these models before discussing some recent developments in plume theory, including an approach which generalizes results obtained separately for the Boussinesq and the non-Boussinesq plume cases. The theory presented - despite its simplicity - has been very successful at describing and explaining the behaviour of plumes across the wide range of scales they are observed. We present solutions to the coupled set of ordinary differential equations (the plume conservation equations) that Morton et al. (1956) derived from the Navier-Stokes equations which govern fluid motion. In order to describe and contrast the bulk behaviour of rising plumes from general area sources, we present closed-form solutions to the plume conservation equations that were achieved by solving for the variation with height of Morton's non-dimensional flux parameter Γ - this single flux parameter gives a unique representation of the behaviour of steady plumes and enables a characterization of the different types of plume. We discuss advantages of solutions in this form before describing extensions to plume theory and suggesting directions for new research. © 2010 The Author. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Delayed decision-making in bistable models

Switching between two modes of operation is a common property of biological systems. In continuous-time differential equation models, this is often realised by bistability, i.e. the existence of two asymptotically stable steadystates. Several biological models are shown to exhibit delayed switching, with a pronounced transient phase, in particular for near-threshold perturbations. This study shows that this delay in switching from one mode to the other in response to a transient input is reflected in local properties of an unstable saddle point, which has a one dimensional unstable manifold with a significantly slower eigenvalue than the stable ones. Thus, the trajectories first approximatively converge to the saddle point, then linger along the saddle's unstable manifold before quickly approaching one of the stable equilibria. ©2010 IEEE.

A PDE viewpoint on basic properties of coordination algorithms with symmetries

Several recent control applications consider the coordination of subsystems through local interaction. Often the interaction has a symmetry in state space, e.g. invariance with respect to a uniform translation of all subsystem values. The present paper shows that in presence of such symmetry, fundamental properties can be highlighted by viewing the distributed system as the discrete approximation of a partial differential equation. An important fact is that the symmetry on the state space differs from the popular spatial invariance property, which is not necessary for the present results. The relevance of the viewpoint is illustrated on two examples: (i) ill-conditioning of interaction matrices in coordination/consensus problems and (ii) the string instability issue. ©2009 IEEE.

A duality principle for homogeneous vectorfields with applications

A duality transformation principle was proposed for converting a positive order homogeneous vectorfield into a negative order homogeneous vectorfield. The principle also converted a uniformly locally asymptotically stable differential equation into a uniformly bounded differential equation. The duality transformations included the geometric framework for homogeneity and the removal of origin from the state space.

Optimization algorithms and ODE's in MDO

There is a need for a stronger theoretical understanding of Multidisciplinary Design Optimization (MDO) within the field. Having developed a differential geometry framework in response to this need, we consider how standard optimization algorithms can be modeled using systems of ordinary differential equations (ODEs) while also reviewing optimization algorithms which have been derived from ODE solution methods. We then use some of the framework's tools to show how our resultant systems of ODEs can be analyzed and their behaviour quantitatively evaluated. In doing so, we demonstrate the power and scope of our differential geometry framework, we provide new tools for analyzing MDO systems and their behaviour, and we suggest hitherto neglected optimization methods which may prove particularly useful within the MDO context. Copyright © 2013 by ASME.

Hybrid numerical methods for exponential models of growth

A series of novel numerical methods for the exponential models of growth are proposed. Based on these methods, hybrid predictor-corrector methods are constructed. The hybrid numerical methods can increase the accuracy and the computing speed obviously, as well as enlarge the stability domain greatly. (c) 2005 Published by Elsevier Inc.

Modelica软件中微分代数方程指标约简算法分析

Modelica建模软件会产生高指标的微分代数方程(DAE),因为现有数值求解器一般不能直接对高指标DAE进行求解,所以Modelica建模软件必须对高指标DAE进行指标约简。为提高Modelica建模软件性能,针对现有指标约简算法,展开了理论分析和实验比较,着重对负权二部图算法进行了详细分析,并给出了负权二部图算法的时间复杂度。理论分析和实验结果表明,负权二部图算法相对Pantelides算法有较好的性能,因此可以考虑在Modelica建模软件中使用负权二部图算法来提高性能。

A Unified Tensor Level Set for Image Segmentation

This paper presents a new region-based unified tensor level set model for image segmentation. This model introduces a three-order tensor to comprehensively depict features of pixels, e.g., gray value and the local geometrical features, such as orientation and gradient, and then, by defining a weighted distance, we generalized the representative region-based level set method from scalar to tensor. The proposed model has four main advantages compared with the traditional representative method as follows. First, involving the Gaussian filter bank, the model is robust against noise, particularly the salt-and pepper-type noise. Second, considering the local geometrical features, e. g., orientation and gradient, the model pays more attention to boundaries and makes the evolving curve stop more easily at the boundary location. Third, due to the unified tensor pixel representation representing the pixels, the model segments images more accurately and naturally. Fourth, based on a weighted distance definition, the model possesses the capacity to cope with data varying from scalar to vector, then to high-order tensor. We apply the proposed method to synthetic, medical, and natural images, and the result suggests that the proposed method is superior to the available representative region-based level set method.

A Relay Level Set Method for Automatic Image Segmentation

This paper presents a new image segmentation method that applies an edge-based level set method in a relay fashion. The proposed method segments an image in a series of nested subregions that are automatically created by shrinking the stabilized curves in their previous subregions. The final result is obtained by combining all boundaries detected in these subregions. The proposed method has the following three advantages: 1) It can be automatically executed without human-computer interactions; 2) it applies the edge-based level set method with relay fashion to detect all boundaries; and 3) it automatically obtains a full segmentation without specifying the number of relays in advance. The comparison experiments illustrate that the proposed method performs better than the representative level set methods, and it can obtain similar or better results compared with other popular segmentation algorithms.

便携式移动机器人姿态解算方法

本文根据便携式移动机器人的特点,采用四元数法解算机器人导航系统的姿态,避免了在机器人运动角度较大时出现奇异点的问题。文中应用改进的四阶龙格-库塔算法解算四元数微分方程,经仿真实验,精度完全能够达到要求。给出了合理的变换公式,在机器人运动范围内,满足了四元数与欧拉角之间转换的一一对应。

基于SVD和能量最小原则的图像自适应降噪算法

基于奇异值分解和能量最小原则,提出了一种自适应图像降噪算法,并给出了基于有界变差的能量降噪模型的代数形式。通过在矩阵范数意义下求能量最小,自适应确定去噪图像重构的奇异值个数。该算法的特点是将能量最小法则和奇异值分解结合起来,在代数空间中建立了一种自适应的图像降噪算法。与基于压缩比和奇异值分解的降噪方法相比,由于该算法避免了图像压缩比函数及其拐点的计算,因此具有快速去噪和简单可行的优点。实验结果证明,该算法是有效的。

软土亚塑性模型与一维非线性固结理论研究

With the great development of Tianjing New Coastal District economy, people need more land to build and live. Land subsidence, which is caused by its special engineering geological conditions, has restricted the further development in the district. Soft soil consolidation is main factor of land subsidence ;thus , on the basis of consolidation theory, the paper make further study on soft soils one-dimension nonlinear consolidation which contains two parts:(1) the nonlinear consolidation of permeability coefficient and compressibility coefficient changing with time and depth, which means real one-dimension nonlinear consolidation;(2) the non-homogeneous consolidation of permeability coefficient and compressibility coefficient only changing with depth. Firstly, nonlinear characteristics of soft soils are elaborated. Hypoplastic theory is introduced to establish a modified soft soils nonlinear constitutive model; the nonlinear governing equation of compressibility coefficient is built, and the nonlinear characteristics of compressibility coefficient are analyzed. Secondly, Considering Load Fluctuation and soil thickness changing ,the consolidation characteristics of single layer is discussed in the paper; meanwhile, on the basis of the Davis and Raymond’s hypothesis and single layer nonlinear consolidation equation, the doubled-layer one-dimension nonlinear consolidation equation is also derived. The solution of the equation is obtained by analytical method, and the consolidation characteristics of doubled-layer soft soil nonlinear theory is also analyzed. Finally, based on assumption that permeability coefficient and compressibility coefficient is varying along depth, single layer soil one-dimension non-homogeneous consolidation differential equation is derived; and the approximate solution is obtained. Furthermore, the single layer non-homogeneous consolidation is extended to double layer non-homogeneous consolidation theory. By using parabolic differential scheme, the matrix equation is established; and the solution of the matrix equation is obtained by chase method. Consolidation characteristics of soil soft single (double) layer non-homogeneous consolidation theory and Terzaghi’s theory are also discussed.

地震波场延拓+菲涅尔体叠加方法及成像效果

In the increasingly enlarged exploration target, deep target layer(especially for the reservoir of lava) is a potential exploration area. As well known, the reflective energy becomes weak because the seismic signals of reflection in deep layer are absorbed and attenuate by upper layer. Caustics and multi-values traveltime in wavefield are aroused by the complexity of stratum. The ratio of signal to noise is not high and the fold numbers are finite(no more than 30). All the factors above affect the validity of conventional processing methods. So the high S/N section of stack can't always be got with the conventional stack methods even if the prestack depth migration is used. So it is inevitable to develop another kind of stack method instead. In the last a few years, the differential solution of wave equation was hold up by the condition of computation. Kirchhoff integral method rose in the initial stages of the ninetieth decade of last century. But there exist severe problems in it, which is are too difficult to resolve, so new method of stack is required for the oil and gas exploration. It is natural to think about upgrading the traditionally physic base of seismic exploration methods and improving those widely used techniques of stack. On the other hand, great progress is depended on the improvement in the wave differential equation prestack depth migration. The algorithm of wavefield continuation in it is utilized. In combination with the wavefield extrapolation and the Fresnel zone stack, new stack method is carried out It is well known that the seismic wavefield observed on surface comes from Fresnel zone physically, and doesn't comes from the same reflection points only. As to the more complex reflection in deep layer, it is difficult to describe the relationship between the reflective interface and the travel time. Extrapolation is used to eliminate caustic and simplify the expression of travel time. So the image quality is enhanced by Fresnel zone stack in target. Based on wave equation, high-frequency ray solution and its character are given to clarify theoretical foundation of the method. The hyperbolic and parabolic travel time of the reflection in layer media are presented in expression of matrix with paraxial ray theory. Because the reflective wave field mainly comes from the Fresnel Zone, thereby the conception of Fresnel Zone is explained. The matrix expression of Fresnel zone and projected Fresnel zone are given in sequence. With geometrical optics, the relationship between object point in model and image point in image space is built for the complex subsurface. The travel time formula of reflective point in the nonuniform media is deduced. Also the formula of reflective segment of zero-offset and nonzero offset section is provided. For convenient application, the interface model of subsurface and curve surface derived from conventional stacks DMO stack and prestack depth migration are analyzed, and the problem of these methods was pointed out in aspects of using data. Arc was put forward to describe the subsurface, thereby the amount of data to stack enlarged in Fresnel Zone. Based on the formula of hyperbolic travel time, the steps of implementation and the flow of Fresnel Zone stack were provided. The computation of three model data shows that the method of Fresnel Zone stack can enhance the signal energy and the ratio of signal to noise effectively. Practical data in Xui Jia Wei Zhi, a area in Daqing oilfield, was processed with this method. The processing results showed that the ability in increasing S/N ratio and enhancing the continuity of weak events as well as confirming the deep configuration of volcanic reservoir is better than others. In deeper target layer, there exists caustic caused by the complex media overburden and the great variation of velocity. Travel time of reflection can't be exactly described by the formula of travel time. Extrapolation is bring forward to resolve the questions above. With the combination of the phase operator and differential operator, extrapolating operator adaptable to the variation of lateral velocity is provided. With this method, seismic records were extrapolated from surface to any different deptlis below. Wave aberration and caustic caused by the inhomogenous layer overburden were eliminated and multi-value curve was transformed into the curve.of single value. The computation of Marmousi shows that it is feasible. Wave field continuation extends the Fresnel Zone stack's application.

Symbolic Integration

SIN and SOLDIER are heuristic programs in LISP which solve symbolic integration problems. SIN (Symbolic INtegrator) solves indefinite integration problems at the difficulty approaching those in the larger integral tables. SIN contains several more methods than are used in the previous symbolic integration program SAINT, and solves most of the problems attempted by SAINT in less than one second. SOLDIER (SOLution of Ordinary Differential Equations Routine) solves first order, first degree ordinary differential equations at the level of a good college sophomore and at an average of about five seconds per problem attempted. The differences in philosophy and operation between SAINT and SIN are described, and suggestions for extending the work presented are made.

Holevo-Ordering and the Continuous-Time Limit for Open Floquet Dynamics

Gough, John, (2004) 'Holevo-Ordering and the Continuous-Time Limit for Open Floquet Dynamics', Letters in Mathematical Physcis 67(3) pp.207-221 RAE2008